Shrinkwrap modifier not applying correctly
About the modifier stack:

When a modifier is applied, it has to be applied as if it is at the top of the modifier stack (the first one), as to achieve the previewed result you will need to apply all other visible modifiers as well.

About the Shrinkwrap issue in particular:

A couple of things could be causing this. You could try:


Make sure your target mesh is high enough resolution (enough faces to deform vertices to)
Use the Offset slider
Use the Keep Above Surface option


Screenshot: